Hyperbolic Geometry ArtworkHyperbolic geometry can be very beautiful. Hyperbolic tilings are not technically fractals, but they appear as fractals when you look at them (because they must be projected into Euclidean space).

The Poincaré ball is the 3D analogy of the Poincaré disk. It is a projection of uniformly tiled hyperbolic polyhedrons from hyperbolic space into Euclidean space. This projection can be generated by recursively applying spherical inversions to the faces of a sphairahedron (a polyhedron with spherical faces). This is a (12,4) Poincaré ball, meaning that is was generated from a hyperbolic dodecahedron having 12 faces with 4 dodecahedrons meeting at each edge. You can purchase this as a poster here.

Hyperbolic polyhedrons can be visualized in Euclidean space using sphairahedrons (polyhedrons with spheres for each face). This picture shows a hyperbolic dodecahedron composed of 12 spherical faces. This image was featured on the cover of McGraw-Hill's 2012 & 2013 Geometry textbooks. Click here to download some POV-Ray code for the hyperbolic dodecahedron.
<< Graphics`Polyhedra`; Show[Graphics3D[Polyhedron[Dodecahedron][[1]]]]

This is what the dodecahedron would look like viewed from the inside with spherical mirrored walls. At certain dihedral angles, this resembles the inside of the Poincaré ball (although not exactly). Notice that when the space becomes elliptic, a black “hole” opens up in the center. This is because the space loops around on itself causing objects beyond the “maximum distance” to appear larger because they are actually closer. Weird huh?

Technically, it is impossible to map a regular Euclidean tiling to the hyperbolic plane but it can still be done with some distortion. This (6,4) tiling of my Dinosaur tessellation was conformally mapped from a hexagon to a hyperbolic hexagon using a Hypergeometric function.

The area inside this circle represents a hyperbolic plane filled with “ideal triangles”. Notice that all the angles inside these triangles go to zero at the edge of the circle. The right animation shows how a single homography can transform the upper half plane into the Poincaré disk.

The Poincaré hyperbolic disk can be conformally mapped to the surface of a sphere. In this mapping, it almost looks like circles are preserved, but not quite. This makes me wonder whether it is possible to conformally map Poincaré disks onto other 3D surfaces. In the left image, the texture was precomputed using forward transformations (this is called the “push forward” method), and then it was mapped to the sphere. In the right image, the the texture was simultaneously calculated using inverse transformations (this is called the “pull back” method) while the sphere was being ray traced. Click here to download a Mathematica notebook for this image. Also, here is some C++ source code for this image.

Click here to download some POV-Ray code for this image. You can also make hyperboloids quickly in POV-Ray using the quadric command:
camera{location <0,10,0> look_at <0,0,0>}
light_source{<0,10,0>,1}
quadric{<1,1,-1>,<0,0,0>,<0,0,0>,1 pigment{rgb 1}}